What is nonelementary integral?

A nonelementary integral is an integral that cannot be expressed in terms of elementary functions that we usually work with, such as exponential, logarithmic, trigonometric, or polynomial functions. These functions can be differentiated and integrated using standard methods, but there are some functions whose integrals cannot be expressed in terms of these elementary functions. Some examples of nonelementary integrals include the Gaussian integral e^(-x^2), the error function erf(x), the Fresnel integral, and the Gamma function.

Finding nonelementary integrals can be a challenging problem in mathematics, and there are several techniques that can be used to evaluate them. Some of these techniques include integration by substitution, integration by parts, trigonometric substitution, partial fractions, and complex analysis. However, even with these techniques, there are some integrals that cannot be evaluated in closed form and can only be approximated numerically using numerical integration techniques.